Mathematical Sciences: Arithmetic Models for Shimura Varieties, L-Functions and Cohomology Groups as Integral Representations

小行星9623269 PI将继续研究以下两个问题：A。利用齐次空间的等变嵌入理论的思想，研究非光滑约化素数上的Shimura簇算术模型的几何和局部性质。 B。研究具有有限群作用的算术簇的上同调中出现的积分表示。这是推广A的整数环的伽罗瓦模结构理论的程序的一部分。Fr; 3 hlich和M. 泰勒到高维算术变种。作者建议调查的不变量描述这些表示与L-函数获得的数据的关系。 这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

9623269 Pappas The PI will continue his work on the following two problems: A. Study the geometry and the local properties of arithmetic models of Shimura varieties over primes of non-smooth reduction, by using ideas from the theory of equivariant embeddings of homogeneous spaces. B. Study the integral representations that appear in the cohomology of arithmetic varieties with finite group actions. This is part of a program for extending the theory of Galois module structure of rings of integers of A. Fr;3hlich and M. Taylor to higher dimensional arithmetic varieties. The author proposes to investigate relations of invariants describing these representations with data obtained from L-functions. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.